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Introduction to Density Functional Theory |
DFT overview |
Density functional theory (DFT) approaches the many-body problem by focusing on the electronic density which is a function of three spatial coordinates instead of finding the wave functions. DFT tries to minimize the energy of a system (ground state) in a self consistent way, and it is very successful in calculating the electronic structure of solid state systems.
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A functional is a function whose argument is itself a function.
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The ground state density
$$n(\textbf{r})$$ determines the external potential energy$$v(\textbf{r})$$ to within a trivial additive constant.
So what Hohenberg-Kohn theorem says, may not sound very trivial. Schrödinger equation says how we can get the wavefunction from a given potential. Once solved the wavefunction (which could be difficult), we can determine the density or any other properties. Now Hohenberg and Kohn theorem says the opposite is also true. For a given density, the potential can be uniquely determined. For non-degenerate ground states, two different Hamiltonian cannot have the same ground-state electron density. It is possible to define the ground-state energy as a function of electronic density.
Total energy of the system
$E(n)$ is minimal when$n(\textbf{r})$ is the actual ground-state density, among all possible electron densities.
The ground state energy can therefore be found by minimizing
The essence of the HK theorem is that the non-degenerate ground-state wave function is a unique functional of the ground-state density:
For any system of
$N$ interacting electrons in a given external potential$v_{ext} (\textbf{r})$ , there is a virtual system of$N$ non-interacting electrons with exactly the same density as the interacting one. The non-interacting electrons subjected to a different external (single particle) potential.
where
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Energy functional is a function of the local charge density:
where
These are a family of functionals that depends on the local density and the local gradient of the density:
There are many flavor of this functional. There are also more advanced functionals: Meta-GGA (e.g., SCAN), hybrids (e.g., B3LYP), nonlocal functionals for van der Waals forces, Grimme's DFT+D (a semi-empirical correction to GGA). They usually produces more accurate result, but computationally more expensive and sometimes numerically unstable.
We can write our Schrödinger in Dirac Bra-Ket notation:
we are going to solve non-interacting single particle Hamiltonian in terms of
known basis functions (plane waves) with unknown coefficients. We start with an
initial guess for the electron density
Self consistency loop in DFT calculation. The above screenshot was taken from lecture slide of Professor Ralph Gevauer from ICTP MAX School 2021.
The potential due to the ions is replaced by the pseudo potentials which removes the oscillations near the atomic core (reducing number of required plane wave basis vectors) and simulates the exact behavior elsewhere. The pseudo potential is also different for different exchange correlation functional, and it is specified in the pseudo potential file. If a system had more than one type of atom, always choose the pseudo potentials with same exchange correlation (e.g., PBE).
It is important to note that DFT is calculations are not exact solution to the
real systems because exact functional (
The wavefunctions are expanded in terms of a basis set. In quantum espresso, the the basis function is plane waves. There exists other DFT codes that use localized basis function as well. Plane waves are simpler but generally requires much large number of them compared to other localized basis sets.
Where
$$ \sum_{\beta} \rm{H}{\alpha\beta} c{i\beta} = \epsilon_i c_{i\alpha} $$
This is a linear algebra problem, solving the above involves diagonalization of
(
Apart from plane waves, various localized basis set could be used, e.g., Linear Combination of Atomic Orbitals (LCAO), Gaussian-type Orbitals (GTO), Linearized Muffin-Tin Orbitals (LMTO). Once could also consider mixed basis sets, such as the Linearized Augmented Plane Waves (LAPW). Localized sets are smaller in size, they can be used for both finite and periodic systems, however they are difficult to use/calculate.
In case of plane waves, we need larger basis set, and requires periodicity. Need to construct supercell for finite systems. Use of pseudopotential reduces the number of required plane waves.
Finding the ground state:
Fourier expansion:
Contribution from higher Fourier components are small, we can limit the sum at
finite
The charge density can be obtained from:
We need two sets of basis vectors: one to store the wavefunctions, and another for the charge density.
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We need about 4 times the cutoff for the charge density compared to the cutoff
for the wavefunction. In case of ultrasoft pseudo potentials, we require a lower
cutoff for energy, therefore ecutrho
might require 8 or 12 times higher than
the ecutwfc
.
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